Using Buchholz's psi notation, the ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\), usually called the "Takeuti-Feferman-Buchholz ordinal", is a large countable ordinal that is the proof-theoretic ordinal of \(\Pi_1^1-\text{CA}+\text{BI}\), a subsystem of second-order arithmetic.

It is the limit of Ferferman's theta notation, as well as the limit of Buchholz's psi notation.

It is also the ordinal measuring the strength of Buchholz hydras with \(\omega\) labels, as well as the upper bound of the SCG function.

It was named by David Madore under the nickname "Gro-Tsen" on wikipedia[1].

Sources Edit

  1. (see the end of "Going beyond the Bachmann-Howard ordinal")

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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